On finite simple groups containing perspectivities

Abstract

PERSPECTIVITIESOne of the most fundamental results in the theory of projective planes is due to R. Baer: Let ~ be a projective plane and i a collineation of ~ of order 2, then either i is a perspectivity or planar.There are almost no general results dealing with collineation groups all of whose involutions are planar. Conversely, if a collineation group contains perspectivities of order 2, then a bunch of general theorems are available. Even if the restriction on the order of a perspectivity is dropped, still better results are known. A milestone among these is the following fundamental theorem due to Chr. Hering [25] : Let G be a finite collineation group acting strongly irreducibly on a projective plane ~. If G contains perspectivities, then with one exception G is an automorphism group of a non-abelian simple group.Geometriae Dedicata 13 (1982) 07-46. 0046-5755/82/0131-007506.00. Proof. The list of maximal subgroups can be found in [44] ; for the remaining statements see [31]. LEMMA 1.3. Let G be isomorphic with Mathieu's group M22. Then the following holds."(a) [GI = 27"32"5"7"11. (b) If M is a maximal subgroup of G, then M ~ La(4), E16 '2~5' E16"A6, AT, E 8 • L3(2), L2(11 ), or A6"2. (c) All involutions in G are conjugate and if t is an involution in G, then C~(t) = E16"24, a subgroup of the maximal subgroup E16'A6.Proof. The list of maximal subgroups can be found in [6], whereas for the remaining statements see [35].

Strongly irreducible collineation groups and the Hall-Janko group

“…We summarise some of the results proved in [9,Appendix]. The notation for elements of ,/2 is that of [1].…”

“…(The exception is when G contains a normal elementary abelian subgroup M of order 9 with CG(M) = M.) ttering's theorem raises the question of which non-abelian simple groups can actual]y occur as M. The case where M is an alternating group or a group of Lie type has been dealt with in various papers by Hering, Walker and Stroth [3], [4], [10]. aeifart and Stroth [9] have considered the case where M is one of the 26 sporadic simple groups and they show that the only group that may occur is the Hall-Janko group ,/2-However, it is not known whether J2 actually does occur. The purpose of this paper is to examine this question.…”

“…Reifart and Stroth in the Appendix to their paper [9] describe some properties of the action of ,/2 on such a finite plane. In Section 2 we establish some further properties.…”

Collineation groups preserving an oval in a projective place of odd order

“…The group G is strongly irreducible on the subplane of II generated by the centres and the axes of involutory homologies in G (see [10,Lemmas 3.3 and 3.5]). So, G?M n or M n by [15].…”

Projective planes with a transitive automorphism group

“…Since we know that an involution is either a perspectivity or a Baer involution, see [8,IV.4.3] it can be assumed in the proofs that all involutions are Baer involutions. There are many papers which consider the action of simple groups on projective planes which make the converse assumption that there are perspectivities, for example, [13,12,11]. There are also papers which consider specific groups acting with very few constraints; see for example [5].…”